Suppose we are in case (I). Note that since \(S_{ij}=A_i \cap A_j\subseteq D_i \cap D_j\) , the points \(p_i(x)\)  and \(p_j(x)\)  are not contained in \(S_{ij}\) . The geodesic \([p_i(x),p_j(x)]\)  either contains a unique endpoint of \(S_{ij}\)  (if \(g_i\)  and \(g_j\)  translate in the same direction along \(S_{ij}\) ) or contains the whole segment \(S_{ij}\)  (if \(g_i\)  and \(g_j\)  translate in opposite directions). In either case, we can choose an endpoint \(s_{ij}\)  of \(S_{ij}\)  such that
\([p_i(x),p_j(x)]\)  is the concatenation of the geodesics \([p_i(x),s_{ij}]\)  and \([s_{ij},p_j(x)]\) . Consider the geodesic quadrilateral with vertices given by \(x, p_i(x),s_{ij}\)  and \(p_j(x)\) . The angle at \(s_{ij}\)  is exactly \(\pi \)  and, by [1]}, the angles at \(p_i(x)\)  and \(p_j(x)\)  are at least \(\frac{\pi }{2}\) . It follows from the Flat Quadrilateral Theorem [1]} that the angle at \(x\)  is zero and therefore (after possibly swapping \(i\)  and \(j\) ) we have that \(p_i(x)\)  lies on the geodesic \([x,p_j(x)]\) . But then, by [1]}, \(p_j(x)=p_j(p_i(x))\in D_j \subseteq A_j \backslash A_j^+\) , which is a contradiction.
